Abstract
We study a stylized vertical distribution channel where a representative manufacturer sells a single kind of good to a representative retailer. The control of the manufacturer is the price discounts, while the control of the retailer is pass-through. In the classical setting, the arising problem is quadratic with respect to wholesale price discount and pass-through. Thus, the optimal sale price is continuous. It seems elegant mathematically but not adequate economically. Therefore we assume that the controls are constant or piece-wise constant. This way, the optimal control problem reduces to the mathematical programming problem where the profit of the manufacturer is quadratic with respect to price discount level(s), while the profit of the retailer is quadratic with respect to pass-through level(s). We study the concavity property of the profits. This allows getting the optimal behavior strategies of the manufacturer and the retailer.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Note that we consider the case when time switches of discount levels are fixed and known. It seems realistic. Indeed, the periods of discounts usually are known. For example, Christmas sales, Winter sales. But, of course, it is possible to consider the situation with non-fixed switches of discount levels.
- 2.
Note that although the profits are quadratic with respect to price discount and pass-through level(s), their concavity is especially important. It is quite appropriate to recall the classic [12]: “Quadratic programming with one negative eigenvalue is NP-hard”.
- 3.
As in [2], \(\gamma >0\) is the sales productivity in terms of motivation, \(\varepsilon >0\) is the discount productivity in terms of motivation. Parameter \(\overline{\alpha }\in [A_1, A_2]\) takes into account the fact that the retailer has some expectations about the wholesale discount: the motivation is reduced if the retailer is dissatisfied with the wholesale discount, i.e., if \(\alpha (t) <\overline{\alpha };\) on the contrary, the motivation increases if \(\alpha (t)> \overline{\alpha }.\) .
- 4.
\(\theta >0\) is the saturation parameter of the market, \(\delta>\) is the retailer’s selling skill, \(\eta >0\) is the discount productivity in terms of sales (the market sensitivity to shelf price discounts).
- 5.
\(\overline{M}>0\) is the initial motivation of the retailer.
- 6.
In particular, three Nash equilibria can be, namely \(\left( \alpha ^{0},\beta ^{0}\right) ,\;\left( \alpha ^{+},\beta ^{+}\right) ,\;\left( \alpha ^{-},\beta ^{-}\right) ,\) where
$$\begin{aligned} \begin{array}{c} \alpha ^0=0, \qquad H\beta ^0+ L=\displaystyle \frac{p K}{q}\, , \\ \alpha ^+=\displaystyle \frac{q(1+\varGamma )}{4p} \, , \quad H\beta ^++ L=\displaystyle \frac{(H+L)(1+\varGamma )}{4}\, , \\ \alpha ^-=\displaystyle \frac{q(1-\varGamma )}{4p} \, , \quad H\beta ^-+ L=\displaystyle \frac{(H+L)(1-\varGamma )}{4}\, , \\ \varGamma =\sqrt{1-\frac{8pK}{q(H+L)}} \ . \end{array} \end{aligned}$$Besides, when the manufacturer is leader, Stackelberg equilibrium can be \((\alpha ^m, \beta ^m),\) where
$$ \alpha ^m=\displaystyle \frac{q(H+L)-pK}{2p(H+L)}= \displaystyle \frac{q\left( 7+\varGamma ^2\right) }{16p} \, , \qquad H\beta ^m+L= \displaystyle \frac{(H+L)\left( 5+3\varGamma ^2\right) }{2\left( 7+\varGamma ^2\right) } \ . $$ - 7.
Note that \( x_{1}\left( \tau _{0}\right) =0\) while \(M_{1}\left( \tau _{0}\right) =\overline{M}\).
- 8.
- 9.
For instance,
$$\begin{aligned} \begin{array}{c} P_{4,2}=s_{1}s_{2}+s_{1}s_{3}+s_{1}s_{4}+s_{2}s_{3}+s_{2}s_{4}+s_{3}s_{4}\\ =s_{1}s_{2}+s_{1}s_{3}+s_{2}s_{3}+\left( s_{1}+s_{2}+s_{3}\right) s_{4}=P_{3,2}+s_{4}P_{3,1}. \end{array} \end{aligned}$$ - 10.
At least, it is easy to get the analogs of (13).
References
Bykadorov, I., Ellero, A., Moretti, E., Vianello, S.: The role of retailer’s performance in optimal wholesale price discount policies. Eur. J. Oper. Res. 194(2), 538–550 (2009)
Bykadorov, I.: Dynamic marketing model: optimization of retailer’s role. Commun. Comput. Inf. Sci. 974, 399–414 (2019)
Bykadorov, I.A., Ellero, A., Moretti, E.: Trade discount policies in the differential games framework. Int. J. Biomed. Soft Comput. Hum. Sci. 18(1), 15–20 (2013)
Nerlove, M., Arrow, K.J.: Optimal advertising policy under dynamic conditions. Economica 29(144), 129–142 (1962)
Bala, P.K.: A data mining model for investigating the impact of promotion in retailing. In: 2009 IEEE International Advance Computing Conference, IACC 2009, 4809092, pp. 670–674 (2009)
Giri, B.C., Bardhan, S.: Coordinating a two-echelon supply chain with price and inventory level dependent demand, time dependent holding cost, and partial backlogging. Int. J. Math. Oper. Res. 8(4), 406–423 (2016)
Giri, B.C., Bardhan, S., Maiti, T.: Coordinating a two-echelon supply chain through different contracts under price and promotional effort-dependent demand. J. Syst. Sci. Syst. Eng. 22(3), 295–318 (2013)
Printezis, A., Burnetas, A.: The effect of discounts on optimal pricing under limited capacity. Int. J. Oper. Res. 10(2), 160–179 (2011)
Routroy, S., Dixit, M., Sunil Kumar, C.V.: Achieving supply chain coordination through lot size based discount. Mater. Today Proc. 2(4–5), 2433–2442 (2015)
Ruteri, J.M., Xu, Q.: The new business model for SMEs food processors based on supply chain contracts. In: International Conference on Management and Service Science, MASS 2011, 5999355 (2011)
Sang, S.: Bargaining in a two echelon supply chain with price and retail service dependent demand. Eng. Lett. 26(1), 181–186 (2018)
Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim. 1(1), 15–22 (1991)
Bykadorov, I.A., Kokovin, S.G., Zhelobodko, E.V.: Product diversity in a vertical distribution channel under monopolistic competition. Autom. Remote Control 75(8), 1503–1524 (2014)
Bykadorov, I., Ellero, A., Funari, S., Kokovin, S., Pudova, M.: Chain store against manufacturers: regulation can mitigate market distortion. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 480–493. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-44914-2_38
Antoshchenkova, I.V., Bykadorov, I.A.: Monopolistic competition model: the impact of technological innovation on equilibrium and social optimality. Autom. Remote Control 78(3), 537–556 (2017)
Bykadorov, I.: Monopolistic competition model with different technological innovation and consumer utility levels. CEUR Workshop Proc. 1987, 108–114 (2017)
Bykadorov, I., Kokovin, S.: Can a larger market foster R&D under monopolistic competition with variable mark-ups? Res. Econ. 71(4), 663–674 (2017)
Bykadorov, I., Gorn, A., Kokovin, S., Zhelobodko, E.: Why are losses from trade unlikely? Econ. Lett. 129, 35–38 (2015)
Acknowledgments
The work was supported in part by the Russian Foundation for Basic Research, projects 18-010-00728 and 19-010-00910, by the program of fundamental scientific researches of the SB RAS, project 0314-2019-0018, and by the Russian Ministry of Science and Education under the 5–100 Excellence Programme.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Bykadorov, I. (2020). Dynamic Marketing Model: The Case of Piece-Wise Constant Pricing. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds) Optimization and Applications. OPTIMA 2019. Communications in Computer and Information Science, vol 1145. Springer, Cham. https://doi.org/10.1007/978-3-030-38603-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-38603-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38602-3
Online ISBN: 978-3-030-38603-0
eBook Packages: Computer ScienceComputer Science (R0)